Ian Stewart

That’s not a typo; there really is an army of cows wandering a maze, at least on the cover. I found only one cow inhabiting a maze in the book proper, but that maze is quite the labyrinth. Still, I’m getting ahead of myself.

If you glance occasionally through Scientific American — even if, like me, you glance through someone else’s copy (thanks, dad!) — then you know that they have for decades run columns on “mathematical games” or “recreational mathematics.” You likely know that the late Martin Gardner started this, and kept it up for decades. One of his successors is Ian Stewart, the author of this text. Those of you familiar with his work there don’t need to read much further, because the twenty-one chapters of this book constitute the third collection of Stewart’s columns. You can guess that this attractive paperback is more convenient than several years’ worth of SciAm magazines, and it might interest you that, for many chapters, Stewart has added reader feedback and given resources for further study. You can now go back to playing with your sphericons (p. 135 of this text).

For the rest of you: Cows in the Maze is a collection of Stewart’s lively columns from Scientific American on “recreational mathematics.” The reader would be right to guess from the armies of cows and engineers on the cover that one chapter describes a maze whose rules change while you navigate it.

Like the cover illustration, the mathematics in this book is playful and invites experiment. Stewart covers a broad range of topics; for starters, probability theory, number theory, knot theory, combinatorial geometry, and mathematical physics. Even algebra shows up in a chapter that explains why you can prove a negative in mathematics; Stewart provides a clear, high-level explanation as to why ruler-and-compass constructions cannot double the cube, trisect the angle, or square the circle.

Topics in this text are user-friendly, and target an audience more general than math majors. Explanations are given, but not proofs. In the chapter on ruler-and-compass constructions, for example, the messy details of Galois theory are left hidden: you tour the palace, but not the plumbing. As such, the text will appeal to math phobes as well as math philes. Most chapters are quite visual and hands-on, encouraging the reader to play with the problem. (See the book’s subtitle: mathematical explorations.) You can throw loaded dice, roll the aforementioned sphericon, explore a calculus for children’s string constructions like Cat’s Cradle, and engage in mathematical dance.

In most chapters, Stewart raises a number of questions along the lines of, “What if…?” or, “How can I…?” Some of these questions are answered; others are left open; and often enough, Stewart supplies both answers and reasons to suggest that the question remains open. Algebraic formulas are more or less absent — It was Gardner, Stewart writes in the introduction, who showed me that mathematics is much broader and richer than anything I’d been exposed to at school — and while Stewart explores deep mathematical ideas, he writes at a level that is accessible not only to the educated, but to the educable. It isn’t light reading, but no sophisticated mathematical background is required to begin. Indeed, many of the problems and solutions appearing throughout the book are due to individuals whose primary career is not mathematical.

The chapters are mostly independent, so you can chew on some pieces at different times. Nor do you have to spend very much time getting to understand them. I have read the whole book and can honestly say that I found two chapters so spectacularly uninteresting that I didn’t bother trying to make sense of them, so I skimmed them instead. Stewart encourages this in the introduction: [D]on’t get hung up on difficult details, plough ahead anyway. Often light then dawns, and if not, you can always go back and try again.

Although Stewart cautions in the introduction that this is not a textbook, it could still prove useful in a classroom or extracurricular setting: an Honors class in mathematics designed primarily for humanities majors, a math club activity, or a Math Circle for high school students. Some of the topics could stimulate ideas for undergraduate or even graduate research projects, and some of it is the result of professional research! Engagement of some of the topics requires movement and construction: go ahead, try to make your way through the maze without cheating. You can have a lot of fun with this book, especially if you see it as the beginning of a lot of new activities that start with the questions, “What if…?” or “How do I…?;”

John Perry is Assistant Professor of Mathematics at the University of Southern Mississippi.